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Klumpenhouwer Network
A Klumpenhouwer Network, named after its inventor, Canadian music theorist and former doctoral student of David Lewin's at Harvard, Henry Klumpenhouwer, is "any network that uses T and/or I operations ( transposition or inversion) to interpret interrelations among pcs" (pitch class sets).Lewin, David (1990). "Klumpenhouwer Networks and Some Isographies That Involve Them", p. 84, ''Music Theory Spectrum'', vol. 12, no. 1 (Spring), pp. 83–120. According to George Perle, "a Klumpenhouwer network is a chord analyzed in terms of its dyadic sums and differences," and "this kind of analysis of triadic combinations was implicit in," his "concept of the cyclic set from the beginning", Perle, George (1993). "Letter from George Perle", ''Music Theory Spectrum'', vol. 15, no. 2 (Autumn), pp. 300–303. cyclic sets being those " sets whose alternate elements unfold complementary cycles of a single interval." "Klumpenhouwer's idea, both simple and profound in its implications, is t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complement (music)
In music theory, ''complement'' refers to either traditional interval complementation, or the aggregate complementation of twelve-tone and serialism. In interval complementation a complement is the interval which, when added to the original interval, spans an octave in total. For example, a major 3rd is the complement of a minor 6th. The complement of any interval is also known as its ''inverse'' or ''inversion''. Note that the octave and the unison are each other's complements and that the tritone is its own complement (though the latter is "re-spelt" as either an augmented fourth or a diminished fifth, depending on the context). In the aggregate complementation of twelve-tone music and serialism the complement of one set of notes from the chromatic scale contains all the ''other'' notes of the scale. For example, A-B-C-D-E-F-G is ''complemented'' by B-C-E-F-A. Note that ''musical set theory'' broadens the definition of both senses somewhat. Interval complementation Rule of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transformation (geometry)
In mathematics, a geometric transformation is any bijection of a set to itself (or to another such set) with some salient geometrical underpinning. More specifically, it is a function whose domain and range are sets of points — most often both \mathbb^2 or both \mathbb^3 — such that the function is bijective so that its inverse exists. The study of geometry may be approached by the study of these transformations. Classifications Geometric transformations can be classified by the dimension of their operand sets (thus distinguishing between, say, planar transformations and spatial transformations). They can also be classified according to the properties they preserve: * Displacements preserve distances and oriented angles (e.g., translations); * Isometries preserve angles and distances (e.g., Euclidean transformations); * Similarities preserve angles and ratios between distances (e.g., resizing); * Affine transformations preserve parallelism (e.g., scaling, shear); * Proje ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isography
An isogloss, also called a heterogloss (see Etymology below), is the geographic boundary of a certain linguistic feature, such as the pronunciation of a vowel, the meaning of a word, or the use of some morphological or syntactic feature. Major dialects are typically demarcated by ''bundles'' of isoglosses, such as the Benrath line that distinguishes High German from the other West Germanic languages and the La Spezia–Rimini Line that divides the Northern Italian languages and Romance languages west of Italy from Central Italian dialects and Romance languages east of Italy. However, an ''individual'' isogloss may or may not have any coterminus with a language border. For example, the front-rounding of /y/ cuts across France and Germany, while the /y/ is absent from Italian and Spanish words that are cognates with the /y/-containing French words. One of the best-known isoglosses is the centum-satem isogloss. Similar to an isogloss, an isograph is a distinguishing feature of a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word isomorphism is derived from the Ancient Greek: ἴσος ''isos'' "equal", and μορφή ''morphe'' "form" or "shape". The interest in isomorphisms lies in the fact that two isomorphic objects have the same properties (excluding further information such as additional structure or names of objects). Thus isomorphic structures cannot be distinguished from the point of view of structure only, and may be identified. In mathematical jargon, one says that two objects are . An automorphism is an isomorphism from a structure to itself. An isomorphism between two structures is a canonical isomorphism (a canonical map that is an isomorphism) if there is only one isomorphism between the two structures (as it is the case for solutions of a univer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vertex (graph Theory)
In discrete mathematics, and more specifically in graph theory, a vertex (plural vertices) or node is the fundamental unit of which graphs are formed: an undirected graph consists of a set of vertices and a set of edges (unordered pairs of vertices), while a directed graph consists of a set of vertices and a set of arcs (ordered pairs of vertices). In a diagram of a graph, a vertex is usually represented by a circle with a label, and an edge is represented by a line or arrow extending from one vertex to another. From the point of view of graph theory, vertices are treated as featureless and indivisible objects, although they may have additional structure depending on the application from which the graph arises; for instance, a semantic network is a graph in which the vertices represent concepts or classes of objects. The two vertices forming an edge are said to be the endpoints of this edge, and the edge is said to be incident to the vertices. A vertex ''w'' is said to be ad ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of Music Theory
The ''Journal of Music Theory'' is a peer-reviewed academic journal specializing in music theory and analysis. It was established by David Kraehenbuehl (Yale University) in 1957. According to its website, " e ''Journal of Music Theory'' fosters conceptual and technical innovations in abstract, systematic musical thought and cultivates the historical study of musical concepts and compositional techniques. The journal publishes research with important and broad applications in the analysis of music and the history of music theory as well as theoretical or metatheoretical work that engages and stimulates ongoing discourse in the field. While remaining true to its original formalist outlook, the journal also addresses the influences of philosophy, mathematics, computer science, cognitive sciences, and anthropology on music theory." The journal is currently edited by Richard Cohn. It has a long and distinguished history of past editors, including Allen Forte Allen, Allen's or Allens ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Recursion
Recursion (adjective: ''recursive'') occurs when a thing is defined in terms of itself or of its type. Recursion is used in a variety of disciplines ranging from linguistics to logic. The most common application of recursion is in mathematics and computer science, where a function being defined is applied within its own definition. While this apparently defines an infinite number of instances (function values), it is often done in such a way that no infinite loop or infinite chain of references ("crock recursion") can occur. Formal definitions In mathematics and computer science, a class of objects or methods exhibits recursive behavior when it can be defined by two properties: * A simple ''base case'' (or cases) — a terminating scenario that does not use recursion to produce an answer * A ''recursive step'' — a set of rules that reduces all successive cases toward the base case. For example, the following is a recursive definition of a person's ''ancestor''. One's ances ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
